Holonomy, symmetry and geometric flows

The geometric objects that can be perceived by our senses are curves and surfaces. Submanifolds provide a natural generalization for higher dimensions of these objects.

The focus of this project is on totally geodesic submanifolds and isoparametric hypersurfaces, intriguing classes of submanifolds with connections to various mathematical areas, often studied using differential geometric, algebraic, or topological methods.The aim of this project is to investigate the interplay of totally geodesic submanifolds with Riemannian holonomy and isoparametric hypersurfaces with certain geometric flows, with the ultimate goal of obtaining results of both intrinsic and extrinsic nature.

Specifically, we intend to complete the classifications of totally geodesic submanifolds in symmetric spaces and of homogeneous hypersurfaces in exceptional symmetric spaces.

We will also use certain classes of isoparametric hypersurfaces in combination with maximum principles to try to prove an Alexandrov-type theorem in the complex hyperbolic space and long-time existence for the hypersymplectic flow.To develop this project, the Experienced Researcher will join the Geometric Analysis team at ULB in Brussels, under the supervision of one of its main researchers, Joel Fine.

The host group has extensive experience in the study of manifolds with special holonomy and geometric flows, using techniques from PDE theory. The training strategy of this project involves assimilating these techniques.

Moreover, the ER has experience in the classical theory of submanifolds in symmetric spaces, as evidenced by his contributions to the field.

The combination of both backgrounds is essential for developing this proposal.Finally, this MSCA fellowship will enhance the convergence of distinct research fields and collaborative networks, generate synergy with the research performed by the Supervisor, diversify the fellow’s mathematical knowledge, and establish him as an independent researcher.